Abstract
The superconducting proximity effect has recently attracted a renewed interest as the basis of topologically nontrivial states in materials with a large spin–orbit interaction, with protected boundary states useful for quantum information technologies. However, spectroscopy of these states is challenging because of the limited control of conventional tunnel barriers. Here we report electronic spectroscopy measurements of the proximity gap in a semiconducting indium arsenide nanowire segment coupled to a superconductor, using quantum dots formed deterministically during the crystal growth. We extract characteristic parameters describing the proximity gap, which is suppressed for lower electron densities and fully developed for larger ones. This gatetunable transition of the proximity effect can be understood as a transition from the long to the short junction regime of subgap bound states in the NW segment. Our device architecture opens up the way to systematic, quantitative spectroscopy studies of subgap states, such as Majoranabound states.
Introduction
Coupling a superconductor (SC) to a metal or a semiconductor results in the socalled superconducting proximity effect in the normal material^{1}. If the proximitized material is a lowdimensional semiconductor, this phenomenon can, for example, be used as a source of spinentangled electrons^{2,3} or superconducting magnetometers^{4}. By combining an swave SC with a onedimensional semiconducting nanowire (NW) with large spin–orbit interaction, one can artificially create a proximity region with superconductivity of pwave character. This can give rise to exotic quantum states at the ends of the SC, such as Majoranabound states^{5,6,7,8}, potentially useful as building blocks for topological quantum information processing^{9,10}. However, a systematic characterization of superconducting subgap states and the proximity region in NWs remains challenging. Several theoretical proposals suggest to use a quantum dot (QD) as a spectrometer to investigate Majoranabound state lifetimes^{11,12}, spin texture^{13}, or parity^{14}. A first experiment was reported recently in which the QD was defined by electrical gating^{7}. However, systematic and deterministic spectroscopy requires a spatially well defined, weakly coupled QD with sharp tunnel barriers across the complete NW, which does not hybridize with the bound states or the SC under investigation. In contrast to earlier spectroscopy experiments in NWs^{15,16}, we use integrated QDs for the tunnel spectroscopy, which separates the spectrometer from the investigated region in a predictable way. To probe the evolution of the proximity effect in a NW segment close to the superconducting contact, we use an insitu grown axial QD in an InAs NW, which homogeneously covers the complete diameter of the NW, in contrast to the standard gatedefined QDs. Compared with previous experiments investigating the spatial distribution and length scales of the superconducting proximity effect in metals^{17,18}, the semiconducting NWs also allow us to control the carrier density and all related parameters, such as the Fermi velocity, in the system. We demonstrate two types of tunnel spectroscopy distinguished by the transport mechanism through the integrated QD, both probing the NW density of states (DOS): the cotunneling regime, where the QD can be seen as a single tunnel barrier, and the sequential tunneling regime, where the QD acts as an energy filter with Coulomb blockade (CB) resonances. The complementary measurements allow us to draw a clear picture of how the proximity gap forms in a NW segment, for which we present an intuitive explanation supported by numerical simulations.
Results
Device and characterization
The QD is defined by two potential barriers that form when the NW crystal structure is changed from zincblende (ZB) to wurtzite (WZ), which can be achieved with atomic precision by controlling the growth parameters^{19,20}. These QDs are electrically and spatially well defined, which allows us to probe the induced gap in the NW segment at a precise distance from the QD and with predictable coupling parameters. Similar insitu grown barriers have been used previously to investigate double QD physics^{21,22}.
We use an InAs NW with an insitugrown axial QD formed by controlling its crystal phase structure in the growth direction^{20}: two thin segments (30 nm) of WZ phase are grown in the otherwise ZB NW. These segments act as atomically precise hardwall tunnel barriers for electrons, because the ZB and WZ band structure align with a conduction band offset of ~100 meV^{23,24}. Consequently, a QD is formed in the ZB section (20 nm) between the two barriers^{23}. A false color scanning electron micrograph of the device discussed here is shown in Fig. 1a, together with a transmission electron micrograph of the atomically precise interface in a representative NW, shown in the inset. The superconducting electrode consists of titanium/aluminum (Ti/Al) and the normal contact of titanium/gold (Ti/Au), fabricated by electron beam lithography. The differential conductance G = dI/dV_{SD} as a function of the backgate voltage V_{BG} in the normal state (in an external magnetic field of B = 50 mT) shows regularly spaced CB resonances (see Supplementary Fig. 1a in Supplementary Note 1) for which we find a systematic increase in the total tunnel coupling Γ with increasing V_{BG} (see Supplementary Fig. 1b in Supplementary Note 1), consistent with the lowering of the tunnel barrier when the band structure is shifted to lower energies with respect to the Fermi energy (see schematic in Supplementary Fig. 2 in Supplementary Note 1)^{23}.
In the device discussed here, the QD is located about L ≈ 350 nm from the Al electrode, leaving a bare NW segment of this length between the QD and the SC, which we refer to as the “lead segment” (LS). This provides a new experimental situation, as the QD is not directly coupled to the SC (as it might be the case in other experiments^{25,26,27}) and can be used to probe the LS. V_{BG} here directly tunes the chemical potential in both the LS and the QD. We record G as a function of the bias voltage V_{SD} applied between the SC and the normal metal contact. If the electrons would tunnel directly from the QD to the superconducting electrode, we would expect to see a gap similar to the one of the bulk value Δ ≈ 210 μeV^{28} of the SC, independent of V_{BG}. As most of the bias drops over the QD, G is proportional to the DOS in the LS. This is the case for positive V_{BG} where carriers accumulate in the NW. Therefore, we can perform spectroscopy on the LS by tunneling from the QD (see Fig. 1b).
An interesting transition in the conductance can already be found in the overview data in Fig. 1c, d presenting regular CB diamonds in the superconducting state (charging energy E_{c} ≈ 6 meV, level spacing ε ≈ 1.5 meV to 2 meV) for two different regimes of V_{BG}. While for positive gate voltages, V_{BG} > 3 V, we observe a superconducting gap around zero bias (c.f. Fig. 1d), for low gate voltages (Fig. 1c) we do not find any features related to superconductivity, but rather the sequence of diamonds expected for the normal state.
The QD can be used as a spectrometer in two different regimes. In the first regime, the QD is kept deep in the CB regime where the charge is fixed for bias voltages in the range of the proximity induced gap Δ*, as Δ* ≪ E_{c}. In this regime, transport is mediated by cotunneling, which is a secondorder process involving the virtual occupation of a QD state^{29}. Here, the QD can be thought of as a single tunnel barrier. In the second regime, the QD electrochemical potential is tuned to a CB resonance, with transport occurring as first order sequential tunneling. In the following, we discuss the experiments in the two regimes one after the other.
Cotunneling spectroscopy
First, we investigate the cotunneling regime, where we can understand the SC–LS as an SN junction, which is weakly coupled to the QD. An example of a measured spectrum is shown in Supplementary Fig. 3a in Supplementary Note 1. It can be characterized by four quantities: the magnitude of the observed gap Δ*, the full width at half the maximum (FWHM) of the peaks at the gap edge, the normal state conductance G_{N} measured at a bias V_{SD} > Δ*/e and G_{S} the conductance at V_{SD} = 0. As the conductance is not fully suppressed at zero bias, we characterize this softness by defining the suppression factor S = G_{S}/G_{N}.
Figure 2a–c show G as a function of V_{BG} and V_{SD} for three different backgate regimes. Crosssections (Fig. 2c: averaged over 20 crosssections in the CB diamond centers) at different V_{BG} in the CB regime are plotted as blue lines. At large gate voltages (V_{BG} ≈ 7 V, Fig. 2a) we find a clear gap around zero bias, which can be detected down to V_{BG} ≈ 2.6 V (Fig. 2b). The same crosssections normalized to G_{N} are shown in Supplementary Fig. 3a in Supplementary Note 1. All crosssections exhibit a suppression of G at V_{SD} = 0 and Δ* ≈ 150 μeV (position of the peak maxima). The FWHM and S are shown to be roughly constant over the investigated gate range, see Supplementary Fig. 3b in Supplementary Note 1, with FWHM ≈ 65 μeV ± 10 μeV and S ≈ 0.5 ± 0.1, respectively. In an ideal SC and weak coupling, S should be close to zero for a strong tunnel barrier. Larger values are often observed for proximityinduced gaps, and referred to as soft gap^{30,31,32}. For lower gate voltages V_{BG} < 2.6 V (Fig. 2b) it is difficult to perform this analysis, because the cotunneling signal is very low compared with the noise floor of the experiment (G ≈ 10^{−5} e^{2}/h). However, we can still observe broad peaks down to V_{BG} ≈ 1.8 V. For V_{BG} < 1.8 V, the differential conductance is too low to observe any signatures of superconductivity with the resolution of our measurement. For even lower gate voltages V_{BG} ≈ 0.2 V (Fig. 2c), no features inside the CB diamonds can be resolved anymore. In summary, in the regime accessible by the cotunneling experiments, the proximity gap characteristics are roughly independent of the gate voltage.
Sequential tunneling spectroscopy
To extract characteristics of the LS for a larger gate range than in the cotunneling regime, we now use the QD resonances (firstorder process through the QD). Panels (i) (“Exp.”) of Fig. 3a–d show detailed measurements of a CB resonance in different backgate regimes, whereas panels (ii) show a simulated map (“model”) discussed below, and panels (iii) show crosssections at V_{BG}, as indicated in panels (i, ii). We will first discuss the experimental results presented in panels (i) of Fig. 3a–d. Afterwards we will compare our findings with the calculated results of a resonant tunneling model, which are presented in panels (ii) of each Fig. 3a–d. In an S–QD–N system, the Coulomb diamond pattern is expected to be affected by superconductivity as presented in Fig. 3e. The tips of the Coulomb diamonds are expected to be shifted by ΔV_{SD} = 2Δ*/e in bias and by ΔV_{BG} = 2Δ*/βe in gate voltage (with β the lever arm of the QD)^{26,33}.
From the QD characteristics in the normal state (see Supplementary Fig. 1a in Supplementary Note 1), we find that the tunnel coupling Γ increases significantly for V_{BG} > 2.5 V. For large Γ, e.g., at V_{BG} ≈ 6.6 V, a CB pattern is shown in Fig. 3a. The CB resonances are broad, but clearly show a suppressed conductance around zero bias. We find a gap of Δ* ≈ 150 μeV, confirming the value obtained in the cotunneling regime (Fig. 2).
A CB resonance with a smaller Γ value at V_{BG} ≈ 2.1 V is shown in Fig. 3b, which is similar to the one in Fig. 3a, but with an additional resolved resonance. As expected (see Fig. 3e), the “CB diamond tips” are shifted in energy by ±Δ^{*}/e and ΔV_{BG} in gate voltage, yielding a consistent value of Δ^{*} ≈ 150 μeV. We observe an additional resonance that crosses through the gap (white arrows). This line corresponds to the alignment of the Fermi levels of the two reservoirs with the QD state. We attribute this resonance to tunneling through the nonzero DOS remaining at zero bias (see position I in Fig. 3e).
At V_{BG} ≈ 0.2 V, the conductance suppression is significantly reduced, as shown in Fig. 3c. The CB diamond tips appear to be only slightly separated and shifted. We note that in this gate range we cannot resolve any signal in the cotunneling spectrum, as discussed in Fig. 2.
At even lower gate voltages, e.g., V_{BG} ≈ 80 mV (Fig. 3d), we do not observe any influence of the superconducting contact, but a regular CB resonance, as found in the normal state.
To extract the characteristic numbers from these data, we use a resonant tunneling model for a S–QD–N junctions. The current is then given by^{26,34} \(I = {\int}_{  \infty }^\infty d ED_{\mathrm{N}}(E) \cdot D_{\mathrm{S}}(E + eV_{{\mathrm{SD}}}) \cdot T_{{\mathrm{QD}}}(E,V_{{\mathrm{BG}}},V_{{\mathrm{SD}}}) \cdot [f_{\mathrm{N}}(E)  f_{\mathrm{S}}(E + eV_{{\mathrm{SD}}})]\), with D_{N}(E) the constant DOS of the normal metal, T_{QD}(E, V_{BG}, V_{SD}) a Lorentzian transmission function, accounting for the resonant tunneling through the QD, and including a broadening due to the finite coupling to the electrodes. f_{S/N} are the Fermi distribution functions for the respective contacts. To account for the softness of the gap, the DOS in the LS can be expressed using the phenomenological Dynes parameter δ by \(D_{\mathrm{S}} = {\mathrm{Re}}(E  i\delta /\sqrt {(E  i\delta )^2  {\mathrm{\Delta }}^{ \ast 2}} )\)^{35}. By adjusting the magnitude of the gap Δ^{*}, the QD resonance broadening Γ and the Dynes parameter δ, we get the conductance maps presented in panel (ii) of each Fig. 3a–d (“model”).
We can reproduce the characteristics of the CB resonance in Fig. 3a using Γ = 150 μeV, which is slightly smaller than what we obtained in the normal state. For the size of the gap we find Δ* = 165 μeV and δ = 0.4 ⋅ Δ* (δ = 65 μeV), resulting in a suppression of S ≈ 0.5, similar to what is found in the cotunneling regime. To reproduce the data in Fig. 3b, we use Γ = 40 μeV (similar to the numbers in the normal state), Δ* ≈ 145 μeV and δ = 0.4 ⋅ Δ*(δ ≈ 60 μeV), i.e., the parameters are almost identical to the ones we obtained in the cotunneling regime at higher gate voltages. The corresponding crosssections agree well with the experiment. We note that the conductance enhancement at the gap edge, as well as the negative differential conductance are well reproduced by the model. We point out that in the sequential tunneling regime, we can extract gap characteristics down to backgate voltages of 0.2 V, which is not possible in the cotunneling regime.
To reproduce the characteristics of the CB resonance in Fig. 3c (V_{BG} ≈ 200 mV), we find that the tunnel coupling is similar to the one in Fig. 3b (Γ = 60 μeV) and δ = 0.5 ⋅ Δ*(δ ≈ 40 μeV). However, the size of the superconducting energy gap is Δ* ≈ 85 μeV, significantly smaller than that found at larger gate voltages. Also here, the model reproduces the data well, illustrated in the corresponding crosssections.
Using the resonant tunneling model to simulate the resonance around V_{BG} ≈ 80 mV (Fig. 3d) we extract Γ = 50 μeV and an upper limit for Δ* of 10 μeV. The model reproduces very well the characteristics of the CB resonance, which essentially corresponds to an N–QD–N device. The resonances in the experiment at higher bias outside the CB are due to excited states, which are not included in the model.
To summarize the measurements in the CB resonance regime (Fig. 3), we observe a transition from a region where the LS acts as a superconducting lead (large gate voltages) with Δ* ≈ 165 μeV to 150 μeV to an intermediate regime with a reduced Δ* ≈ 85 μeV, to a regime without effects of superconductivity. The parameters extracted from the resonant tunneling model demonstrate a clear evolution of Δ* in the LS.
Discussion
The evolution of the induced gap Δ* as a function of V_{BG} is summarized in Fig. 4a. The curve shows a sharp transition from a clearly resolved energy gap for V_{BG} > 0 to a fully suppressed gap at V_{BG} < 0.
Although the observed proximity feature can be well fitted by a broadened BCS DOS, this approach is not an adequate description, as there are only few states in the quasi onedimensional NW lead. Qualitatively, one can understand the transition by considering only a few modes in the LS. All electrons at energies within the gap of the SC are Andreev reflected (AR) at the SC, giving rise to Andreevbound states (ABSs).
We interpret the observed transition in Δ* qualitatively as a gatetunable transition of ABSs forming in the LS from the long to the short junction limit. Both limits are defined by comparing the physical length of the junction, L, with the characteristic length scale \(L_{\mathrm{c}} = \hbar v_{\mathrm{F}}/\widetilde \Delta\), which is often called coherence length in the ballistic limit^{36}. Assuming a rather low Fermi velocity (v_{F} = 2 × 10^{5} m/s), we find L_{c} ≈ 850 nm. Here v_{F} is the Fermi velocity in the LS, ℏ the Planck constant, and \(\widetilde \Delta\) the proximity gap induced by the aluminum contact in the NW directly below the SC (see inset of Fig. 4a).
In the short junction limit (L ≪ L_{c}) the energy of the ABSs (E_{ABS}) in the LS is dominated by the phase change due to AR at the SC/LS interface (dashed line in schematic), where \(\widetilde \Delta\) is assumed to change abruptly. In this limit, the energies of the ABSs are “pushed” to \(E_{{\mathrm{ABS}}} \cong \widetilde \Delta\), resulting in a superconducting proximity gap in the LS similar to \(\widetilde \Delta\), i.e., \(\Delta ^ \ast \sim \widetilde {\mathrm{\Delta }}\)^{37}. In the long junction limit (L ≫ L_{c}), E_{ABS} is determined by the phase acquired in the LS, which scales with k_{F} = m* ⋅ v_{F}/ℏ ∝ v_{F}, with m* the effective electron mass. In this limit, E_{ABS} can take on smaller values, thus filling the proximity gap^{36}. We therefore can tune E_{ABS} by tuning v_{F} and the electron density in the LS using V_{BG}. For very positive gate voltages, E_{F} is relatively far up in the conduction band with a correspondingly large v_{F}, resulting in a large L_{c}, bringing the LS to the short junction limit, L < L_{c}. In contrast, when we align the bottom of the conduction band to E_{F} by lowering V_{BG}, v_{F} and L_{c} are strongly reduced, bringing the LS into the long junction limit, L > L_{c}, and \(E_{{\mathrm{ABS}}} < \widetilde {\mathrm{\Delta }}\). As a result, the apparent gap Δ* is reduced by the ABSs moving into the gap. We note that \(\widetilde \Delta\) in the NW segment below the SC is screened by the SC and is therefore not gate tunable.
To support this qualitative picture, we employ a numerical model, in which we combine the Green’s function method with a tight binding model. The properties of the SC are taken into account as a selfenergy dressing the bare Green’s function in a NW section below S, where a superconducting gap \(\widetilde \Delta\) is induced depending only on the coupling to S (details can be found in the Supplementary Note 2). This region is coupled to a bare NW segment (of length L = 350 nm) modeling the LS (see inset of Fig. 4a). We then investigate the local DOS at a distance L = 350 nm as a function of the gate voltage in the NW (Fig. 4b). Like for the experimental data, we extract Δ* as the distance between the maxima in the DOS. Here the number of ABSs as well as the ABS energy is determined by the length of the LS. The resulting Δ* of this calculation is plotted as a function of V_{BG} as a red line in Fig. 4a. In excellent agreement with the experiment, we observe a sharp transition in the detected gap, corresponding to the transition from the short to the long junction limit. In the model, Δ* tends to zero when E_{F} is aligned to the bottom of the conduction band (V_{BG} ≈ −0.2 V), just before the LS is fully depleted and v_{F} → 0. For a better understanding we study the dependence of an individual ABS as a function of V_{BG} in Supplementary Fig. 4 in the Supplementary Note 3. Evolving from the long to the short junction limit, the states with energy E < Δ* move toward the gap, resulting in a fully formed smooth proximity gap Δ*. We note that in this model the width of the ABS is a free parameter, which we set to 25 μeV. In seeming contradiction to the depleted LS, the experiment still shows several CB resonances below this gate voltage, which we attribute to evanescent modes from the NW segment below S (which is not depleted) that couples weakly to the QD wavefunctions, rendering the QD couplings highly asymmetric with very low transmission amplitudes, as observed in the experiment.
The gap in our devices is soft, i.e., the conductance suppression at zero bias is significantly lower than in NWs with an epitaxial Al shell^{38,39}. The reason for this remains unclear and is a priori not expected from the model. The evaporated bulk Al shows a hard gap (S ≈ 0.01) when used in standard, largearea metallic S–I–S tunnel junctions measured in a similar experimental setup (see Supplementary Fig. 5 in Supplementary Note 4). Introducing random spatial potential fluctuations at the NW–S interface^{40,41} in the presented numerical model does not account for the observed small suppression either (see Supplementary Fig. 6 in Supplementary Note 5). However, we cannot exclude other types of disorder or nonideal material interfaces as the origin of the “soft gap”. In the ABS picture, the softness of the gap is determined by the life time of the ABSs in the LS. This broadening can have different physical origins, namely (1) tunneling to the QD, which should exhibit a similar tunability as the QD life time, (2) single particle tunneling to the NW segment below S, e.g., by an inverse proximity effect due to the gold nanoparticle used for the NW growth, and (3) quasiparticle excitation by microwave radiation absorption that might be different in NW devices compared to metallic ones.
In conclusion, we present a systematic study of the apparent transport gap in a NW segment induced by a proximity coupled SC. For this purpose we introduce QD tunnel spectroscopy enabled by insitugrown axial tunnel barriers. We observe a gatetunable transition of the gap amplitude from a fully developed, constant proximity gap at large electron densities to smaller values and ultimately a complete suppression of the gap at low densities. The data are consistent with a transition from the short junction limit of an SN device to the long junction limit with ABSs forming at energies also below the gap energy. This transition occurs when the Fermi energy is close to the bottom of the conduction band and the respective Fermi velocity tends toward zero. Our experiments demonstrate that NWs with insitugrown barriers, in our example with crystal phase engineered barriers, are very useful to perform transport spectroscopy in SC–semiconductor hybrid systems. We have thus introduced a novel spectroscopy tool, which is well suited to study superconducting bound states in semiconducting NWs, and might be useful to tackle fundamental limitations found in recent studies of Majoranabound states^{42,43}. The next steps toward investigating Majorana devices on this platform include technological progress, e.g., the possibility to contact the NW epitaxially with a SC as well as precise alignment of this contact close to the insitugrown QD.
Methods
Fabrication and measurement techniques
The InAs NWs were grown by metalorganicvaporphase epitaxy and have an average diameter of 70 nm. The two segments of WZ crystal phase forming the tunnel barriers have a thickness of 30 nm. The zincblende segment in between, which defines the QD, has a width of 25 nm.
The NWs were transferred mechanically from the growth substrate to a degenerately pdoped silicon substrate with a SiO_{2} capping layer (400 nm). The substrate is used as a global back gate. For the electron beam lithography we employ predefined markers and contact pads made of Ti/Au (5 nm /45 nm). The normal metal contact to the NW is made of Ti/Au (5 nm/70 nm) and the superconducting contact of Ti/Al (5 nm/80 nm). Before each evaporation step, the native oxide of the NW is removed by an Argon ion sputtering.
All measurements were carried out in a dilution refrigerator at a base temperature of 20 mK. Differential conductance has been measured using standard lockin techniques (V_{ac} = 10 μV, f_{ac} = 278 Hz).
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work has received funding from the Swiss National Science Foundation, the Swiss Nanoscience Institute, and the Swiss NCCR QSIT. C.S. has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program: grant agreement 787414, ERCAdv TopSupra. D.C. received funding from the European Unions Horizon 2020 research and innovation program (ERC Starting Grant, agreement number 757725). S.L., M.N., K.A.D. and C.T. acknowledge financial support by the Knut and Alice Wallenberg Foundation (KAW) and the Swedish Research Council (VR). We thank C. Reeg and S. Hoffman for fruitful discussions.
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C.J. fabricated the devices and performed the measurements together with R.D. C.J., R.D. and A.B. analyzed the data. A.B. provided the resonant tunneling model. D.C. provided the numerical calculations. S.L., M.N., K.A.D. and C.T. developed the nanowire structure. C.S. and A.B. planned and designed the experiments, and participated in all discussions. All authors contributed to the manuscript.
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Jünger, C., Baumgartner, A., Delagrange, R. et al. Spectroscopy of the superconducting proximity effect in nanowires using integrated quantum dots. Commun Phys 2, 76 (2019). https://doi.org/10.1038/s4200501901624
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